Transport and Mixing in Complex and Turbulent FlowsProgress since the 2010 IMA workshop
Jean-Luc Thiffeault
Department of MathematicsUniversity of Wisconsin – Madison
Institute for Mathematics and its ApplicationsMinneapolis, MN, 9 May 2014
Supported by NSF
1 / 38
http://www.math.wisc.edu/~jeanluchttp://www.math.wisc.eduhttp://www.wisc.eduhttp://www.wisc.edu
Three personal examples
To be able to prepare in time, I decided to narrow the focus to a fewpapers written at the IMA, with several participants:
Lin, Z., Doering, C. R., & Thiffeault, J.-L. (2011a). J. Fluid Mech. 675, 465–476
Thomases, B., Shelley, M., & Thiffeault, J.-L. (2011). Physica D, 240,1602–1614
Lin, Z., Thiffeault, J.-L., & Childress, S. (2011b). J. Fluid Mech. 669, 167–177
Zhi George Lin was a postdoc here and is now at Zhejiang University.
I’ll discuss the papers and some of their impact.
I apologize for the shameless focus on my papers. . .
2 / 38
Optimization of mixing
Inspired by ‘mix-norm’ of Mathew et al. (2005): H−1/2 → H−1.
J. Fluid Mech. (2011), vol. 675, pp. 465–476. c© Cambridge University Press 2011doi:10.1017/S0022112011000292
465
Optimal stirring strategies for passivescalar mixing
ZHI LIN1, JEAN-LUC THIFFEAULT2
AND CHARLES R. DOERING3†1Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, USA
2Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA3Department of Mathematics, Department of Physics and Center for the Study of Complex Systems,
University of Michigan, Ann Arbor, MI 48109, USA
(Received 4 September 2010; revised 30 October 2010; accepted 12 January 2011;
first published online 10 March 2011)
We address the challenge of optimal incompressible stirring to mix an initiallyinhomogeneous distribution of passive tracers. As a quantitative measure of mixingwe adopt the H −1 norm of the scalar fluctuation field, equivalent to the (square rootof the) variance of a low-pass filtered image of the tracer concentration field. Firstwe establish that this is a useful gauge even in the absence of molecular diffusion:its vanishing as t → ∞ is evidence of the stirring flow’s mixing properties in the senseof ergodic theory. Then we derive absolute limits on the total amount of mixing,as a function of time, on a periodic spatial domain with a prescribed instantaneousstirring energy or stirring power budget. We subsequently determine the flow fieldthat instantaneously maximizes the decay of this mixing measure – when such aflow exists. When no such ‘steepest descent’ flow exists (a possible but non-genericsituation), we determine the flow that maximizes the growth rate of the H −1 norm’sdecay rate. This local-in-time optimal stirring strategy is implemented numericallyon a benchmark problem and compared to an optimal control approach using arestricted set of flows. Some significant challenges for analysis are outlined.
Key words: mathematical foundations, mixing, nonlinear dynamical systems
1. IntroductionThe enhancement of mixing by stirring in incompressible flows is an important
phenomenon in a wide variety of applications in sciences and engineering. A naturalquestion is: how efficient a mixer can an incompressible flow be? This fundamentalquestion, more precisely posed, is the subject of this paper.
In principle, given an appropriate quantitative measure of mixing along with suitableconstraints on the accessible class of flow fields, the most efficient mixing strategy maybe determined by solving an optimal control problem. In practice this may be difficult,so it is useful to consider other approaches that might more easily be implemented, atleast theoretically or computationally. Moreover, it is always useful to know absolutelimits on how fast mixing could ever be achieved subject to the relevant constraints.Such bounds provide a scale upon which particular strategies may be evaluated togauge their effectiveness. Here we propose and analyse a theoretical scenario with a
† Email address for correspondence: [email protected]
3 / 38
Direct descendant of IMA
JOURNAL OF MATHEMATICAL PHYSICS 53, 115611 (2012)
Optimal mixing and optimal stirring for fixed energy,fixed power, or fixed palenstrophy flows
Evelyn Lunasin,1 Zhi Lin,2,a) Alexei Novikov,3 Anna Mazzucato,3
and Charles R. Doering41Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, USA2Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310013,People’s Republic of China3Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania16802, USA4Department of Mathematics, Department of Physics, and Center for the Study of ComplexSystems, University of Michigan, Ann Arbor, Michigan 48109, USA
(Received 31 March 2012; accepted 19 July 2012; published online 5 October 2012)
We consider passive scalar mixing by a prescribed divergence-free velocity vectorfield in a periodic box and address the following question: Starting from a giveninitial inhomogeneous distribution of passive tracers, and given a certain energybudget, power budget, or finite palenstrophy budget, what incompressible flow fieldbest mixes the scalar quantity? We focus on the optimal stirring strategy recentlyproposed by Lin et al. [“Optimal stirring strategies for passive scalar mixing,” J.Fluid Mech. 675, 465 (2011)] that determines the flow field that instantaneouslymaximizes the depletion of the H− 1 mix-norm. In this work, we bridge some of thegap between the best available a priori analysis and simulation results. After recallingsome previous analysis, we present an explicit example demonstrating finite-timeperfect mixing with a finite energy constraint on the stirring flow. On the other hand,using a recent result by Wirosoetisno et al. [“Long time stability of a classical efficientscheme for two dimensional Navier-Stokes equations,” SIAM J. Numer. Anal. 50(1),126–150 (2012)] we establish that the H− 1 mix-norm decays at most exponentiallyin time if the two-dimensional incompressible flow is constrained to have constantpalenstrophy. Finite-time perfect mixing is thus ruled out when too much cost isincurred by small scale structures in the stirring. Direct numerical simulations in twodimensions suggest the impossibility of finite-time perfect mixing for flows with fixedpower constraint and we conjecture an exponential lower bound on the H− 1 mix-norm in this case. We also discuss some related problems from other areas of analysisthat are similarly suggestive of an exponential lower bound for the H− 1 mix-norm.C© 2012 American Institute of Physics. [http://dx.doi.org/10.1137/110834901]
Dedicated to Peter Constantin on the occasion of his 60th birthday.
I. INTRODUCTION
The advection of a substance by an incompressible flow is important in many physical settings.This process often involves complex evolving structures of wide range of space and time scales.Here, we concentrate on the case of scalar advection where the transported quantity is passive, sohas negligible feedback on the flow. Given a stirring velocity flow field u(x, t) with ∇ · u = 0, weconsider the advection of a passive scalar field ρ(x, t) by a smooth incompressible flow field u(x, t)
0022-2488/2012/53(11)/115611/15/$30.00 C©2012 American Institute of Physics53, 115611-1
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.104.46.196 On: Tue, 06 May 2014 14:17:39
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Checkerboard maps provide nice bounds
Lunasin et al. (2012)5 / 38
H−1 led to renewed interest in bounds on mixing
(Two workshop participants with a student.)
LOWER BOUNDS ON THE MIX NORM OF PASSIVE SCALARS
ADVECTED BY INCOMPRESSIBLE
ENSTROPHY-CONSTRAINED FLOWS.
GAUTAM IYER, ALEXANDER KISELEV, AND XIAOQIAN XU
Abstract. Consider a diffusion-free passive scalar θ being mixed by an in-compressible flow u on the torus Td. Our aim is to study how well this scalarcan be mixed under an enstrophy constraint on the advecting velocity field.
Our main result shows that the mix-norm (‖θ(t)‖H−1 ) is bounded below byan exponential function of time. The exponential decay rate we obtain is not
universal and depends on the size of the support of the initial data. We also
perform numerical simulations and confirm that the numerically observed de-
cay rate scales similarly to the rigorous lower bound, at least for a significant
initial period of time. The main idea behind our proof is to use recent work of
Crippa and DeLellis (’08) making progress towards the resolution of Bressan’s
rearrangement cost conjecture.
1. Introduction
The mixing of tracer particles by fluid flows is ubiquitous in nature, and haveapplications ranging from weather forecasting to food processing. An importantquestion that has attracted attention recently is to study “how well” tracers can bemixed under a constraint on the advecting velocity field, and what is the optimalchoice of the “best mixing” velocity field (see [24] for a recent review).
Our aim in this paper is to study how well passive tracers can be mixed underan enstrophy constraint on the advecting fluid. By passive, we mean that thetracers provide no feedback to the advecting velocity field. Further, we assumethat diffusion of the tracer particles is weak and can be neglected on the relevanttime scales. Mathematically, the density of such tracers (known as passive scalars)is modeled by the transport equation
(1.1) ∂tθ(x, t) + u · ∇θ = 0, θ(x, 0) = θ0(x).To model stirring, the advecting velocity field u is assumed to be incompressible.For simplicity we study (1.1) with periodic boundary conditions (with period 1),mean zero initial data, and assume that all functions in question are smooth.
The first step is to quantify “how well” a passive scalar is mixed in our context.For diffusive passive scalars, the decay of the variance is a commonly used measureof mixing (see for instance [10,14,22,25] and references there in). But for diffusionfree scalars the variance is a conserved and does not change with time. Thus,
This material is based upon work partially supported by the National Science Foundation un-
der grants DMS-1007914, DMS-1104415, DMS-1159133, DMS-1252912. GI acknowledges partial
support from an Alfred P. Sloan research fellowship. AK acknowledges partial support from a
Guggenheim fellowship. The authors also thank the Center for Nonlinear Analysis (NSF Grants
No. DMS-0405343 and DMS-0635983), where part of this research was carried out.
1
arX
iv:1
310.
2986
v3 [
mat
h.A
P] 1
2 Fe
b 20
14
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More H−1. . .
IOP PUBLISHING NONLINEARITY
Nonlinearity 26 (2013) 3279–3289 doi:10.1088/0951-7715/26/12/3279
Maximal mixing by incompressible fluid flows
Christian Seis
Department of Mathematics, University of Toronto, 40 St. George Street, M5S 2E4, Toronto,Ontario, Canada
Received 4 September 2013, in final form 7 September 2013Published 21 November 2013Online at stacks.iop.org/Non/26/3279
Recommended by B Eckhardt
AbstractWe consider a model for mixing binary viscous fluids under an incompressibleflow. We prove the impossibility of perfect mixing in finite time for flows withfinite viscous dissipation. As measures of mixedness we consider a Monge–Kantorovich–Rubinstein transportation distance and, more classically, the H−1
norm. We derive rigorous a priori lower bounds on these mixing norms whichshow that mixing cannot proceed faster than exponentially in time. The rate ofthe exponential decay is uniform in the initial data.
Mathematics Subject Classification: 76D55,76F25
1. Introduction
The present manuscript is concerned with optimal stirring strategies for binary mixtures ofincompressible viscous fluids. More precisely, we study decay rates of certain mixing normswith respect to a constrained velocity field. We focus on passive scalar mixing, which meansthat the feedback of the transported quantity on the flow field is negligible. To model thebinary mixture, we consider an indicator function ρ = ρ(t, x) which takes the values +1 and−1 only, so that the sets {ρ = 1} and {ρ = −1} represent the regions in which the fluidconsists of component ‘A’ and component ‘B’, respectively. As usual, t and x are the time andspace variable, respectively. The stirring velocity field will be denoted by u = u(t, x), and weassume this vector field to be smooth. The transport of the passive scalar by the incompressibleflow is then described by the system
∂tρ + u · ∇ρ = 0, (1)∇ · u = 0, (2)
and we impose the initial condition ρ(0, x) = ρ0(x) ∈ {±1}. For mathematical convenience,we finally assume that all quantities are periodic in the spatial variables with period cell [0, 1)d .Observe that the total mass of each species is preserved under the flow. In the case of a critical
0951-7715/13/123279+11$33.00 © 2013 IOP Publishing Ltd & London Mathematical Society Printed in the UK & the USA 3279
7 / 38
Eventually, a review paper
IOP PUBLISHING NONLINEARITY
Nonlinearity 25 (2012) R1–R44 doi:10.1088/0951-7715/25/2/R1
INVITED ARTICLE
Using multiscale norms to quantify mixing andtransport
Jean-Luc Thiffeault
Department of Mathematics, University of Wisconsin–Madison, Madison, WI, USA
E-mail: [email protected]
Received 6 September 2005, in final form 4 October 2011Published 20 January 2012Online at stacks.iop.org/Non/25/R1
Recommended by B Eckhardt
AbstractMixing is relevant to many areas of science and engineering, including thepharmaceutical and food industries, oceanography, atmospheric sciences andcivil engineering. In all these situations one goal is to quantify and often then toimprove the degree of homogenization of a substance being stirred, referred toas a passive scalar or tracer. A classical measure of mixing is the variance of theconcentration of the scalar, which is the L2 norm of a mean-zero concentrationfield. Recently, other norms have been used to quantify mixing, in particular themix-norm as well as negative Sobolev norms. These norms have the advantagethat unlike variance they decay even in the absence of diffusion, and their decaycorresponds to the flow being mixing in the sense of ergodic theory. GeneralSobolev norms weigh scalar gradients differently, and are known as multiscalenorms for mixing. We review the applications of such norms to mixing andtransport, and show how they can be used to optimize the stirring and mixingof a decaying passive scalar. We then review recent work on the less-studiedcase of a continuously replenished scalar field—the source–sink problem. Inthat case the flows that optimally reduce the norms are associated with transportrather than mixing: they push sources onto sinks, and vice versa.
Mathematics Subject Classification: 76R50, 37A25, 46E35, 65K10
1. Introduction
One of the most vexing questions about fluid mixing is how to measure it. People typicallyknow it when they see it, but specific applications require customized measures. For example, ameasure might be too fine-grained for some applications that do not require thorough mixing.Some measures, such as residence time distributions, are designed for open-flow situationswhere fluid particles are only stirred for a certain amount of time. Others, such as the rigorous
0951-7715/12/020001+44$33.00 © 2012 IOP Publishing Ltd & London Mathematical Society Printed in the UK & the USA R1
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Mixing in viscoelastic flows
Follow-up papers have focused on the flow, not mixing per se.
Author's personal copy
Physica D 240 (2011) 1602–1614
Contents lists available at SciVerse ScienceDirect
Physica D
journal homepage: www.elsevier.com/locate/physd
A Stokesian viscoelastic flow: Transition to oscillations and mixingBecca Thomases a,∗, Michael Shelley b, Jean-Luc Thiffeault ca Department of Mathematics, University of California, Davis, CA 95616, United Statesb Courant Institute of Mathematical Sciences, New York University, New York City, NY 10012, United Statesc Department of Mathematics, University of Wisconsin Madison, WI 53706, United States
a r t i c l e i n f o
Article history:Available online 25 June 2011
Keywords:ViscoelasticityInstabilityMixingMicrofluidics
a b s t r a c t
To understand observations of low Reynolds number mixing and flow transitions in viscoelastic fluids,we study numerically the dynamics of the Oldroyd-B viscoelastic fluid model. The fluid is driven by asimple time-independent forcing that, in the absence of viscoelastic stresses, creates a cellular flow withextensional stagnation points. We find that at O(1) Weissenberg number, these flows lose their slavingto the forcing geometry of the background force, become oscillatory with multiple frequencies, and showcontinual formation and destruction of small-scale vortices. This drives flow mixing, the details of whichwe closely examine. These new flow states are dominated by a single-quadrant vortex, which may bestationary or cycle persistently from cell to cell.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
In the past several years, it has come to be appreciated thatin low Reynolds number flow the nonlinearities provided bynon-Newtonian stresses of a complex fluid can provide richdynamical behaviors more commonly associated with highReynolds number Newtonian flow. For example, experimentsby Steinberg and collaborators have shown that dilute polymersuspensions being sheared in simple flow geometries can exhibithighly time-dependent dynamics and efficient mixing [1–3]. Thecorresponding experiments using Newtonian fluids do not – andindeed cannot – show such nontrivial dynamics. One importantconstraint on the dynamics of a Stokesian Newtonian fluid isreversibility [4], which is lost when the fluid is viscoelastic [5,6].
Both mixing and irreversibility are complex phenomena buteven the understanding of elastic instabilities in viscoelastic fluidsis incomplete. Elastic instabilities in low Reynolds number fluids,where inertia is negligible, have been studied extensively for sometime; see [7–14]. Elastic instabilities are observed at low ormodestflow rates where inertial forces are negligible but elastic forces arestrong, and have been linked to the creation of secondary vortexflows [15] and increased flow resistance [16].
Extensional flows, such as the flow in a four-roll mill or flowin a cross-channel, can be more effective in locally stretchingand aligning polymers than a standard shear flow [17]. Asthe macroscopic flow depends on the microscopically generated
∗ Corresponding author. Tel.: +1 5305542988; fax: +1 5307526635.E-mail address: [email protected] (B. Thomases).
stresses, a flow in an extensional geometry may exhibit aninstability more readily than a flow in a shearing geometry. Thismay be due to the fact that a shear flow can be decomposedinto an extensional flow and a rotational flow and the vorticityin the fluid tends to rotate the fluid microstructure away fromthe principal axes of stretching [18,13]. Experiments have shownthat polymer molecules are strongly stretched as they pass nearextensional points in amicro-channel cross flow [19,20]. Schroederet al. [19] visualized single-molecule stretching and bistability atstagnation points. In the work of Arratia et al. [20], molecularstretching is inferred and two flow instabilities, dependent onthe flow strain rate, are demonstrated. After the onset of thefirst instability, the flow becomes deformed and asymmetric butremains steady; at higher strain rates the velocity field fluctuatesin time and can produce mixing. The first transition appears to bea forward bifurcation to a bistable steady state; see also [21,22].In [23], (henceforth TS2009) these instabilities are demonstratednumerically for a 2D periodic flow, and these results are discussedin greater detail here. Xi and Graham [24] also found numericallyan oscillatory instability for sufficiently largeWeissenberg numberin an extensional flow geometry, and they suggest a possiblemechanism for the instability due to the concentration of stressnear the extensional point in the flow. In [25], Berti et al. shownumerically that flows with a 2D periodic shearing force can giverise to non-stationary dynamics.
In this paper, we study computationally a viscoelastic fluidin an extensional flow. As our flow model, we use the Oldroyd-B equations with polymer stress diffusion in the zero Reynoldsnumber (Stokes) limit. The Stokes–Oldroyd-B model is attractiveas it arises from a simple conception of the microscopic origin ofviscoelasticity [26,27]. The bulk fluid is composed of a Newtonian
0167-2789/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2011.06.011
9 / 38
Biomixing
J. Fluid Mech. (2011), vol. 669, pp. 167–177. c© Cambridge University Press 2011doi:10.1017/S002211201000563X
167
Stirring by squirmers
ZHI LIN1, JEAN-LUC THIFFEAULT1,2† ANDSTEPHEN CHILDRESS3
1Institute for Mathematics and Applications, University of Minnesota – Twin Cities,207 Church Street SE, Minneapolis, MN 55455, USA
2Department of Mathematics, University of Wisconsin – Madison, 480 Lincoln Drive,Madison, WI 53706, USA
3Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street,New York, NY 10012, USA
(Received 10 July 2010; revised 20 October 2010; accepted 22 October 2010;
first published online 1 February 2011)
We analyse a simple ‘Stokesian squirmer’ model for the enhanced mixing due toswimming micro-organisms. The model is based on a calculation of Thiffeault &Childress (Phys. Lett. A, vol. 374, 2010, p. 3487), where fluid particle displacementsdue to inviscid swimmers are added to produce an effective diffusivity. Here we showthat, for the viscous case, the swimmers cannot be assumed to swim an infinitedistance, even though their total mass displacement is finite. Instead, the largestcontributions to particle displacement, and hence to mixing, arise from randomchanges of direction of swimming and are dominated by the far-field stresslet termin our simple model. We validate the results by numerical simulation. We alsocalculate non-zero Reynolds number corrections to the effective diffusivity. Finally,we show that displacements due to randomly swimming squirmers exhibit probabilitydistribution functions with exponential tails and a short-time superdiffusive regime,as found previously by several authors. In our case, the exponential tails are due to‘sticking’ near the stagnation points on the squirmer’s surface.
Key words: mixing, micro-organism dynamics
1. IntroductionSwimming creatures affect their environment in many ways, and one which has
received attention recently is how they mix the surrounding fluid. This phenomenonis called biogenic mixing or biomixing. The most striking and controversial setting isthe ocean: Dewar et al. (2006) suggested that marine life might have an impact onvertical mixing in the ocean.
Katija & Dabiri (2009) proposed that the dominant effect involved in biomixing isthe mass displacement due to a swimming body. This phenomenon is called Darwiniandrift, after Darwin (1953), though the displacement due to a moving cylinder wasobtained by Maxwell (1869). Thiffeault & Childress (2010) derived the effectivediffusivity of an ‘ideal gas’ of randomly distributed non-interacting swimmers, andshowed that it depends on the induced squared displacement of fluid particles by theswimmers, as opposed to the net mass displaced.
† Email address for correspondence: [email protected]
10 / 38
Munk’s Idea
Though it had been mentioned earlier, the first to seriously consider therole of ocean biomixing was Walter Munk (1966):
“. . . I have attempted, without much success, to interpret [the eddydiffusivity] from a variety of viewpoints: from mixing along the oceanboundaries, from thermodynamic and biological processes, and frominternal tides.”
11 / 38
Ocean biomixing: Basic observations
The idea lay dormant for almost 40 years; then
• Huntley & Zhou (2004) analyzed swimming of 100 (!) species,ranging from bacteria to blue whales. Typical turbulent energyproduction is ∼ 10−5 W kg−1. Total is comparable to energydissipation by major storms.
• Another estimate comes from the solar energy captured: 63 TeraW,something like 1% of which ends up as mechanical energy (Dewaret al., 2006).
• Kunze et al. (2006) find that turbulence levels during the day in aninlet were 2 to 3 orders of magnitude greater than at night, due toswimming krill.
• However, Kunze has failed to find this effect again on subsequentcruises. Visser (2007) has questioned whether small-scale turbulencecan lead to overturning.
12 / 38
In situ experiments
Katija & Dabiri (2009) looked at jellyfish:
play movie (Palau’s Jellyfish Lake.) Correct length scale is path length?13 / 38
http://www.math.wisc.edu/~jeanluc/movies/Katija2009_nature08207-s4.mpghttp://en.wikipedia.org/wiki/Jellyfish_Lake
Displacement by a moving body
Maxwell (1869); Darwin (1953); Eames et al. (1994)
14 / 38
A sequence of kicks
Inspired by Einstein’s theory of dif-fusion (Einstein, 1956): a test particleinitially at x(0) = 0 undergoes Nencounters with an axially-symmetricswimming body:
x(t) =N∑
k=1
∆L(ak , bk) r̂k
∆L(a, b) is the displacement, ak , bkare impact parameters, and r̂k is a di-rection vector.
L
a
target particle
swimmerb
�
(a > 0, but b can have ei-
ther sign.)
15 / 38
After squaring and averaging, assuming isotropy:〈|x|2〉
= N〈∆2L(a, b)
〉where a and b are treated as random variables with densities
dA/V = 2da db/V (2D) or 2πa da db/V (3D)
Replace average by integral:〈|x|2〉
=N
V
∫∆2L(a, b)dA
Writing n = 1/V for the number density (there is only one swimmer)and N = Ut/L (L/U is the time between steps):
〈|x|2〉
=Unt
L
∫∆2L(a, b) dA
16 / 38
Effective diffusivity
Putting this together,〈|x|2〉
=2Unt
L
∫∆2L(a, b) da db = 4κt, 2D
〈|x|2〉
=2πUnt
L
∫∆2L(a, b)a da db = 6κt, 3D
which defines the effective diffusivity κ.
If the number density is low (nLd � 1), then encounters are rare and wecan use this formula for a collection of particles.
17 / 38
A ‘gas’ of swimmers
−L/2 0 L/2−L/2
0
L/2
x
y
−1.5 −1 −0.5 0 0.5−5
−4
−3
−2
−1
0
1
start
end
x
y
play movie 100 cylinders, box size = 1000
18 / 38
http://www.math.wisc.edu/~jeanluc/movies/cylinder_gas.mp4
How well does the dilute theory work?
0 20 40 60 80 1000
50
100
150
200
t
〈|x|2〉/2nU`3
n = 10−3
n = 5 × 10−4
n = 10−4
theory
19 / 38
Squirmers: The viscous world
Considerable literature on transport due to microorganisms: Wu & Libchaber(2000); Hernandez-Ortiz et al. (2005); Saintillian & Shelley (2007); Ishikawa & Pedley (2007);
Underhill et al. (2008); Ishikawa (2009); Leptos et al. (2009)
Lighthill (1952), Blake (1971), and more recently Ishikawa et al. (2006)have considered squirmers:
• Sphere in Stokes flow;• Steady velocity specified
at surface, to mimiccilia;
• Steady swimmingcondition imposed (nonet force on fluid).
(Drescher et al., 2009) (Ishikawa et al., 2006)
20 / 38
Particle motion for squirmer
A particle near the squirmer’s swimming axisinitially (blue) moves towards the squirmer.
After the squirmer has passed the particle fol-lows in the squirmer’s wake.
(The squirmer moves from bottom to top.)
play movie
21 / 38
http://www.math.wisc.edu/~jeanluc/movies/squirmer_flyby.avi
Squirmers: Trajectories
The two peaks in the displacement plot come from ‘incomplete’trajectories:
b/L = 0 b/L = 0.5 b/L = 1
For long path length, the effective diffusivity is independent of theswimming path length, and yet the dominant contribution arises from thefiniteness of the path (uncorrelated turning directions).
22 / 38
Squirmers: Transport
0 50 100 150 2000
1
2
3
4
5
6
x 10−4
t
〈|x|2 〉
23 / 38
Some recent papers on biomixing I
Doostmohammadia, A., Stocker, R., & Ardekani, A. M. (2011). Proc. Natl. Acad. Sci. USA,109 (10), 3856–3861
Eckhardt, B. & Zammert, S. (2012). Eur. Phys. J. E, 35, 96
Kunze, E. (2011). J. Mar. Res. 69 (4-6), 591–601
Jepson, A., Martinez, V. A., Schwarz-Linek, J., Morozov, A., & Poon, W. C. K. (2013). Phys.Rev. E, 88, 041002
Katija, K. (2012). J. Exp. Biol. 215, 1040–1049
Khurana, N., Blawzdziewicz, J., & Ouellette, N. T. (2011). Phys. Rev. Lett. 106, 198104
Kurtuldu, H., Guasto, J. S., Johnson, K. A., & Gollub, J. P. (2011). Proc. Natl. Acad. Sci.USA, 108 (26), 10391–10395
Lambert, R. A., Picano, F., Breugem, W.-P., & Brandt, L. (2013). J. Fluid Mech. 733, 528–557
Leshansky, A. M. & Pismen, L. M. (2010). Phys. Rev. E, 82, 025301
Miño, G. L., Dunstan, J., Rousselet, A., Clément, E., & Soto, R. (2013). J. Fluid Mech. 729,423–444
Morozov, A. & Marenduzzo, D. (2014). Soft Matter, 10, 2748–2758
Noss, C. & Lorke, A. (2014). Limnol. Oceanogr. 59 (3), 724–732
Parra-Rojas, C. & Soto, R. (2013). Phys. Rev. E, 87, 053022
Parra-Rojas, C. & Soto, R. (2014). arXiv:1404.4857
24 / 38
Some recent papers on biomixing II
Pushkin, D. O., Shum, H., & Yeomans, J. M. (2013). J. Fluid Mech. 726, 5–25
Pushkin, D. O. & Yeomans, J. M. (2013). Phys. Rev. Lett. 111, 188101
Pushkin, D. O. & Yeomans, J. M. (2014). arxiv:1403.2619
Rousseau, S., Kunze, E., Dewey, R., Bartlett, K., & Dower, J. (2010). J. Phys. Ocean. 40 (9),2107–2121
Saintillan, D. (2010). Physics, 3, 84
Saintillan, D. & Shelley, M. J. (2012). J. Roy. Soc. Interface, 9, 571–585
25 / 38
New effects are being included
In particular, Pushkin and Yeomans confirm that in many cases transportis dominated by entrainment rather than drift:
Dentr ∼1
2dnU`Ventr
Fluid Mixing by Curved Trajectories of Microswimmers
Dmitri O. Pushkin* and Julia M. Yeomans
The Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, United Kingdom(Received 23 July 2013; published 31 October 2013)
We consider the tracer diffusion Drr that arises from the run-and-tumble motion of low Reynolds
number swimmers, such as bacteria. Assuming a dilute suspension, where the bacteria move in
uncorrelated runs of length �, we obtain an exact expression for Drr for dipolar swimmers in three
dimensions, hence explaining the surprising result that this is independent of �. We compare Drr to the
contribution to tracer diffusion from entrainment.
DOI: 10.1103/PhysRevLett.111.188101 PACS numbers: 47.63.Gd, 64.70.pv, 82.70.�y, 87.16.Uv
As microswimmers, such as bacteria, algae, or activecolloids, move they produce long-range velocity fieldswhich stir the surrounding fluid. As a result particles andbiofilaments suspended in the fluid diffuse more quickly,thus helping to ensure an enhanced nutrient supply.Following the early studies of mixing in concentratedmicroswimmer suspensions [1–3], recent experimentshave demonstrated enhanced tracer diffusion in dilute sus-pensions of Chlamydomonas reinhardtii, Escherichia coli,and self-propelled particles [4–8]. Simulations have foundsimilar behavior [9–11] and microfluidic devices exploit-ing the enhanced transport due to motile organisms havebeen suggested [12,13]. However, a theoretical descriptionof fluctuations and tracer mixing in active systems remainsa challenge even for very dilute suspensions of micro-swimmers. The statistics of fluid velocity fluctuationswas studied in [14–16]. As the tracer displacements atshort times are proportional to fluid velocities, these resultscharacterize the short-time statistics of tracer displace-ments. In particular, the fluid velocity fluctuations turnout, generically, non-Gaussian. Features of the long-timetracer displacement statistics remain unknown.
The Reynolds number associated with bacterial swim-ming is �10�4–10�6. Therefore, the flow fields that resultfrom the motion obey the Stokes equations and the farvelocity field can be described by a multipole expansion.The leading order term in this expansion, the Stokeslet (orOseen tensor), which decays with distance �r�1, is theflow field resulting from a point force acting on the fluid.However biological swimmers, which are usually suffi-ciently small that gravity can be neglected, move autono-mously and therefore have no resultant force or torqueacting upon them. Hence, the Stokeslet term is zero andthe flow field produced by the microswimmers containsonly higher order multipoles, for example, dipolar contri-butions, �1=r2, and quadrupolar terms, �1=r3.
The absence of the Stokeslet term has important reper-cussions for the way in which tracer particles are advectedby swimmers. The angular dependences of the dipolarvelocity field—shown in Fig. 1—and of higher order multi-poles of the flow field lead to looplike tracer trajectories.
For a distant swimmer, moving along an infinite straighttrajectory these loops are closed [17] and would not lead toenhanced tracer diffusion.The paths of bacteria or active colloids are, however, far
from infinite straight lines. For example, periodic tumbling(abrupt and substantial changes in direction) is a well-established mechanism by which microorganisms such asE. coli can move preferentially along chemical gradients.Even in the absence of tumbling, microswimmers typicallyhave curved paths due to rotational diffusion or nonsym-metric swimming strokes. For noninfinite swimmer trajec-tories tracers no longer move in closed loops and theswimmer reorientations cause enhanced diffusion [11]. Inthis Letter we obtain an exact expression for the diffusionconstant Drr due to uncorrelated random reorientations ofdipolar swimmers in three dimensions. The result allows usto explain the surprising observation [11] that, for dipolarswimmers in 3D, the diffusivity is independent of theswimmer run length. We then extend our results to givescaling arguments for Drr for swimmers confined to gen-eral dimensions d (but retaining the 3D nature of the flowfield) and a flow field that decays as r�m where, forexample, m ¼ 2 corresponds to dipolar swimmers such
(b)
(c)
(d)
FIG. 1 (color online). (a) The angular dependence of thedipolar flow field. The velocity decays as r�2, where r is thedistance from the swimmer. (b) A typical, closed-loop tracertrajectory for an infinite, straight swimmer path and tracervelocity� swimmer velocity. (c) A typical trajectory for a finiteswimmer path. (d) A typical entrained trajectory, for an infiniteswimmer path, and tracer close to the swimmer [17].
PRL 111, 188101 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending
1 NOVEMBER 2013
0031-9007=13=111(18)=188101(5) 188101-1 � 2013 American Physical Society
26 / 38
Some dampened enthusiasm for ocean biomixing
Journal of Marine Research, 69, 591–601, 2011
Fluid mixing by swimming organisms in thelow-Reynolds-number limit
by Eric Kunze1
ABSTRACTRecent publications in the fluid physics literature have suggested that low-Reynolds-number swim-
ming organisms might contribute significantly to ocean mixing. These papers have focussed on themass transport due to fluid capture and disturbance by settling or swimming particles based on classi-cal fluid mechanics flows but have neglected the role of molecular property diffusion. Scale-analysisof the property conservation equation finds that, while properties with low molecular diffusivities canhave enhanced mixing for typical volume fractions in aggregations of migrating zooplankton, thismixing is still well below that due to internal-wave breaking so unlikely to be important in the ocean.
1. Introduction
Recent interest has been sparked into whether the motile ocean biosphere can contributesignificantly to ocean mixing. Energetic arguments (Munk, 1966; Dewar et al., 2006) sug-gest that up to 1 TW might be available while scale-analysis indicates that aggregations ofswimming marine organisms ranging in size from O(1-cm) krill to O(1-m) cetaceans (Hunt-ley and Zhou, 2004) might be able to generate high-Reynolds-number turbulent kineticenergy dissipation rates ε ∼ O(10−5 W kg−1). Early observational support for such num-bers (Kunze et al., 2006) was not borne out by subsequent more extensive microstructuremeasurements which found that at least 90% of the time either (i) dissipation rates ε werenot significantly higher during dawn and dusk migrations of acoustic backscattering layers(Rippeth et al., 2007), (ii) dissipation rates were elevated in aggregations but two orders ofmagnitude below the predictions of Huntley and Zhou (Rousseau et al., 2010; Lorke andProbst, 2010), or (iii) though dissipation rates were elevated, mixing efficiencies Γ werevery low (Gregg and Horne, 2009).
Another line of research suggests that swimming or settling particles could induce signif-icant mixing without generating turbulence by dragging captured fluid impelled by inertialor viscous forces behind them. These studies have invoked Darwin’s (1953) drift flow asa starting point although this flow was shown to be ill-defined by Eames et al. (1994).Both idealized low- and high-Reynolds-number flows have been considered. Katija and
1. Applied Physics Laboratory, University of Washington, Seattle, Washington, 98105, U.S.A. email:[email protected]
591
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Recent experiments back Kunze
Direct observation of biomixing by vertically migrating zooplankton
Christian Noss * and Andreas Lorke
Institute for Environmental Sciences, University of Koblenz-Landau, Landau, Germany
Abstract
The potential contribution of swimming zooplankton to the vertical mixing of stratified waters has been thetopic of an ongoing scientific debate. Current estimates, which are primarily based on scale analyses andnumerical simulations, range from negligible effects to significant contributions that are comparable in magnitudeto physical driving forces, such as wind and tides. Here, we analyzed laboratory observations of fluid mixing thatare caused by vertically migrating zooplankton (Daphnia magna) in a density-stratified water column. Mixingrates were quantified at the scale of individual organisms in terms of the dissipation rates of small-scale spatialvariance of tracer concentration measured by laser-induced fluorescence. At the bulk scale, we analyzed temporalchanges in the mean density stratification. Organism and bulk scale observations were used to estimate apparentdiffusion coefficients in trails of individuals and organism groups. Mean diffusivities of 0.8–5.1 3 1029 m2 s21,which were averaged over trail volumes of 1.5–13 3 1025 m3, are on the same order of magnitude as the moleculardiffusivity of salt. A comparable diffusivity (1.1 3 1029 m2 s21) was estimated on the bulk scale, and the initialdensity stratification, although frequently passed by migrating Daphnia, was preserved over the 5 d experimentalperiod. The present results agree with scaling arguments and suggest the negligible enhancement of verticaltransport in comparison with the turbulent mixing that is typically observed in oceans and lakes.
The contribution of the biosphere to large-scale verticaltransport and to the mixing of heat and solutes in stratifiedoceans and lakes has been the topic of ongoing scientificdebate. Dewar et al. (2006) estimated a global rate at whichmechanical energy is produced by swimming animals of1012 W, and Huntley and Zhou (2004) estimated a rate onthe order of magnitude O , 1025 W kg21 within animalaggregations. Therefore, biologically induced productionrates of kinetic energy are on the same order of magnitudeas those kinetic energy production rates that are caused bymajor winds and tides and also can be expected tocontribute significantly to mixing and to diapycnaltransport. A first observation in a dense swarm of krill(Kunze 2011) supports these estimations. Although furtherfield measurements in animal aggregations revealed elevat-ed dissipation rates of kinetic energy (Lorke and Probst2010; Rousseau et al. 2010), these measurements were twoorders of magnitude below the magnitude that waspredicted by Huntley and Zhou (2004) or revealed a lowmixing efficiency (Gregg and Horne 2009).
However, experimental observations (Kunze et al. 2006;Lorke and Probst 2010; Rousseau et al. 2010) predomi-nantly focused on the turbulent kinetic energy dissipationrates around somewhat larger organisms and withinaggregations (O , 1 cm to O , 1 m), which are associatedwith higher Reynolds number flows. Additionally, abun-dant, but small, organisms, such as zooplankton, poten-tially contribute to mixing and transport because theseorganisms often cross regions of strong vertical gradients(e.g., the thermocline during diel vertical migration). Visser(2007) argued that the mixing efficiency of small (O # 1 cm)organisms is low due to the small size of the producedhydrodynamic disturbances. Thus far, all referencedestimations and measurements consider turbulence to be
the major source for mixing, where the mixing efficiency islimited by the size of the overturning length scale, which isassumed to be similar to organism size. These consider-ations neglect fluid transport by fluid drift (Katija 2012).
The mixing due to drift is also related to the swimmingmode (Jiang and Strickler 2007) and could be equallyefficient for all sizes (Katija and Dabiri 2009) or evenhigher for low–Reynolds number swimmers (Thiffeault andChildress 2010). Experimental observations of the driftbehind a swimming jellyfish (Katija and Dabiri 2009), aswell as numerical simulations (Dabiri 2010; Thiffeault andChildress 2010), showed that fluid can be displaced overdistances that are much larger than the organism body size.The apparent efficiency of simulated transport and mixing,however, strongly depends on the applied model assump-tions and boundary conditions. Dabiri (2010) estimateddiffusivities that were three orders of magnitudes higherthan the molecular diffusivity of heat for passively movingindividuals in dense aggregations using numerical simula-tions of the velocity field in a stratified inviscid fluid.Following Leshansky and Pismen (2010), transport andmixing will be less efficient if the organisms are self-propelled (i.e., described as force dipoles).
Kunze (2011) criticized the application of the ideal flowtheory for modeling transport and mixing by smallorganisms. First, viscous boundary layers would injectvorticity into their wakes and, hence, develop intoturbulence at high Reynolds numbers. Second, moleculardiffusion in the cross-flow direction potentially short-circuits the shear dispersion, even if viscosity enhancesthe drift volume at low Reynolds numbers. Using scaleanalyses, Kunze (2011) found enhanced eddy diffusivitiescaused by aggregated migrating zooplankton that were,however, negligible compared with typical diffusivities inthe ocean (e.g., caused by internal wave breaking).Nevertheless, a central presumption of this scale analysis* Corresponding author: [email protected]
Limnol. Oceanogr., 59(3), 2014, 724–732
E 2014, by the Association for the Sciences of Limnology and Oceanography, Inc.doi:10.4319/lo.2014.59.3.0724
724
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More refinements of the model
Enhanced diffusion of tracer particles in dilutebacterial suspensions
Alexander Morozov* and Davide Marenduzzo
Swimming bacteria create long-range velocity fields that stir a large volume of fluid and move around
passive particles dispersed in the fluid. Recent experiments and simulations have shown that long-time
mean-squared displacement of passive particles in a bath of swimming bacteria exhibits diffusive
behaviour with an effective diffusion coefficient significantly larger than its thermal counterpart. A
comprehensive theoretical prediction of this effective diffusion coefficient and the understanding of the
enhancement mechanism remain a challenge. Here, we adapt the kinetic theory by Lin et al., J. Fluid
Mech., 2011, 669, 167 developed for ‘squirmers’ to the bacterial case to quantitatively predict enhanced
diffusivity of tracer particles in dilute two- and three-dimensional suspensions of swimming bacteria. We
demonstrate that the effective diffusion coefficient is a product of the bacterial number density, their
swimming speed, a geometric factor characterising the velocity field created by a single bacterium, and a
numerical factor. We show that the numerical factor is, in fact, a rather strong function of the system
parameters, most notably the run length of the bacteria, and that these dependencies have to be taken
into account to quantitatively predict the enhanced diffusivity. We perform molecular-dynamics-type
simulations to confirm the conclusions of the kinetic theory. Our results are in good agreement with the
values of enhanced diffusivity measured in recent two- and three-dimensional experiments.
1 Introduction
Recent interest in suspensions of self-propelled colloidalparticles stems from their relevance to a variety of disciplines.1
In physics, they provide one of the simplest models to under-stand statistical mechanics of out-of-equilibrium systems2 andhydrodynamics and rheology of active matter.3,4 In biology, themotility of bacteria and eukaryotic microorganisms is linked tothe understanding of various diseases,5 fertility6 and biomixingin oceans.7 In engineering, it has been demonstrated thatmotile particles can be made to perform work8–10 and delivercargo.11
Bacteria are one of the most readily available realisations ofself-propelled particles. Their individual motility and collectivebehaviour have been extensively studied.12,13 Many speciespropel by pushing the surrounding uid backwards by rotatinglong thin agella. The propulsive force applied to the uid isthen compensated by the drag the uid exerts on the bacterium.Thus, locally, bacteria act as self-propelled force-dipoles thatstir the uid in a large volume around them. The long-rangedvelocity elds created by bacteria result in an inducedmotion ofpassive particles suspended in the uid such as dead bacteria,nutrients, small droplets of other uids, etc. This so-calledenhanced diffusion is potentially relevant for inducing feedingcurrents around microorganisms and biomixing in oceans.7
A systematic study of enhanced diffusion started with thepioneering work by Wu and Libchaber,14 who measured theeffective diffusion coefficient of large colloidal particles in anE. coli suspension in a quasi-2D free standing soap lm. Wu andLibchaber14 concluded that at long times colloidal particlesbehaved diffusively with the effective diffusion coefficient beingabout 100 times larger than the thermal one. Since then manystudies have conrmed similar behaviour. Long-time diffusivebehaviour of tracers was observed in dilute suspensions ofE. coli,15–20 B. subtilis,21 alga Chlamydomonas reinhardtii,22,23 andsynthetic swimmers,17 with experiments performed in quasi-2Dthin lms14,17,19,21,23 or 3D geometries.15,16,18,20,22 These studiesemployed either colloidal particles or non-motile bacteria astracers, both comparable in size with the swimmers, with theexception of the work by Kim and Breuer,15 who considereddiffusion of small Dextranmolecules in a bath of E. coli bacteria.On the theoretical side, simulations of tracers with self-propelled particles of various types conrm diffusive behav-iour24–26 with a diffusion coefficient signicantly larger than itsequilibrium value in the absence of swimmers.
Experiments,17,19,20,22 theory,19,20,24,26–29 and simulations25,26
provide evidence that the enhanced diffusion coefficient scaleslinearly with the so-called active ux: the product of the numberdensity of swimmers n and their swimming speed U. In order toobtain a quantity of the same dimension as the diffusion coef-cient, the active ux should be multiplied by a length scale tothe fourth power. The precise understanding of the origin ofthis length scale and the value of the numerical prefactor in the
SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh, UK.
E-mail: [email protected]
Cite this: Soft Matter, 2014, 10, 2748
Received 15th August 2013Accepted 6th January 2014
DOI: 10.1039/c3sm52201f
www.rsc.org/softmatter
2748 | Soft Matter, 2014, 10, 2748–2758 This journal is © The Royal Society of Chemistry 2014
Soft Matter
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Chlamydomonas reinhardtii
play movie
[Guasto, J. S., Johnson, K. A., & Gollub, J. P. (2010). Phys. Rev. Lett. 105, 168102]
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http://www.math.wisc.edu/~jeanluc/movies/Guasto2010_start.mp4
Probability density of displacements
Non-Gaussian PDF with ‘exponential’ tails:
[Leptos, K. C., Guasto, J. S., Gollub, J. P., Pesci, A. I., & Goldstein, R. E. (2009).
Phys. Rev. Lett. 103, 198103]
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Probability density of displacements
Leptos et al. (2009) claim a reasonable fit of their PDF with the form
P∆t(∆x) =1− f√2π δg
e−(∆x)2/2δ2g +
f
2δee−|∆x |/δe
They observe the scalings δg ∼ Ag (∆t)1/2 and δe ∼ Ae(∆t)1/2, where Agand Ae depend on φ.
They call this a diffusive scaling, since ∆x ∼ ∆t1/2. Their point is thatthis is strange, since the distribution is not Gaussian.
Commonly observed in diffusive processes that are a combination oftrapped and hopping dynamics (Wang et al., 2012).
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PDF: Theory vs experiment
−4 −2 0 2 410
−4
10−3
10−2
10−1
100
x
ρX
t(x)
The normalized PDF for experimental data (dashed) agrees well withsimple swimmer models. Eckhardt & Zammert (2012) have aphenomenological model.
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Summary
H−1 as a measure of mixing:
• Allows analytical progress (bounds).• Related to ‘Bressan conjecture.’
Viscoelastic mixing:
• A fairly open area. . .• Unfortunately has not caught on yet.
Biomixing:
• Settled in the ocean case?• Interest has shifted to microswimmers.• Many more lab experiments.• We understand the details better (probabilistic tools).• The approach developed at the IMA has been used and refined by
others. Compares well to data and numerical experiments.
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BiomixingDilute theorySimulationsSquirmersConclusionsReferences
